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update Ch13-Recursion
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‎notebooks/Ch13-Recursion.ipynb‎

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@@ -82,7 +82,7 @@
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"metadata": {},
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"outputs": [],
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"source": [
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"sys.setrecursionlimit(1000000)"
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"sys.setrecursionlimit(1_000_000)"
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]
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},
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{
@@ -164,135 +164,83 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Factorial definition\n",
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"### Sum of First N positive integers\n",
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"\n",
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"```\n",
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" 1! = 1 (base case)\n",
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" n! = n.(n-1)! for n > 1 (general case)\n",
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"```\n",
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"- use recursion to find the sum of first N positive integers\n",
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"\n",
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"- Exercise - Implement factorial recursive solution"
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"```\n",
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" first_sum(1) = 1 (base case)\n",
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" first_sum(n) = n + first_sum(n-1) for n > 1 (general case)\n",
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"```"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"def factorial(n):\n",
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"def first_sum(n):\n",
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" if n == 1:\n",
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" return 1\n",
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" else:\n",
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" return n*factorial(n-1)"
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" return n + first_sum(n-1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"factorial(100)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [],
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"source": [
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"from timeit import timeit"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.032029791036620736"
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"5050"
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]
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},
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"execution_count": 15,
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# run it \n",
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"timeit('factorial(10_000)', globals=globals(), number=1)"
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"first_sum(100)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"# use tail recursion optimization\n",
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"# Python doesn't optimize tail recursion \n",
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"# read the blog by Guido van Rossum - http://neopythonic.blogspot.com/2009/04/tail-recursion-elimination.html\n",
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"def factorial_tail(n, running_tot=1):\n",
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"def sum_tail(n, running_tot=1):\n",
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" if n == 1:\n",
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" return running_tot\n",
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" else:\n",
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" return factorial_tail(n-1, running_tot*n)"
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" return sum_tail(n-1, running_tot+n)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000"
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]
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},
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"execution_count": 12,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"factorial_tail(100, 1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.07616704609245062"
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"5050"
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]
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},
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"execution_count": 14,
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# in fact it takes longer for Python as tail recurion is not optimized\n",
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"# stack usage is O(n)\n",
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"timeit('factorial_tail(10_000, 1)', globals=globals(), number=1)"
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"sum_tail(100, 1)"
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]
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},
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{
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"execution_count": null,
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"metadata": {},
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"source": [
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"execution_count": null,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"2178309"
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]
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},
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"execution_count": 17,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"outputs": [],
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"source": [
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"fib(32)\n",
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"#print(count)\n",
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.5792814260348678"
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]
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},
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"execution_count": 18,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"outputs": [],
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"source": [
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"timeit(\"fib(32)\", globals=globals(), number=1)"
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]
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},
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"cell_type": "code",
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"execution_count": null,
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"2178309\n"
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]
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}
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],
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"outputs": [],
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"source": [
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"print(fib_tail(32, 0, 1))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 24,
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"execution_count": null,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"1.3248994946479797e-05"
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]
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},
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"execution_count": 24,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"outputs": [],
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"source": [
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"timeit(\"fib_tail(32, 0, 1)\", globals=globals(), number=1)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Factorial definition\n",
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"\n",
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"```\n",
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" 1! = 1 (base case)\n",
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" n! = n * (n-1)! for n > 1 (general case)\n",
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"```\n",
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"\n",
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"- Exercise - Implement factorial recursive solution; see homework assignment!"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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},
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"cell_type": "code",
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"execution_count": 3,
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"execution_count": null,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Requirement already up-to-date: pip in /Users/rbasnet/miniconda3/lib/python3.7/site-packages (19.2.1)\n",
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"Requirement already satisfied: pygame in /Users/rbasnet/miniconda3/lib/python3.7/site-packages (1.9.4)\n"
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]
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}
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],
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"outputs": [],
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"source": [
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"%%bash\n",
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"pip install --upgrade pip\n",
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},
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"pygame 1.9.4\n",
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"Hello from the pygame community. https://www.pygame.org/contribute.html\n"
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]
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}
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],
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"outputs": [],
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"source": [
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"# animated fractal; when done just close the window or force kill if not responding!\n",
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"# this program doesn't run in colab or online services. You must run locally from notebook or as a script!\n",
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},
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"cell_type": "code",
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},
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"cell_type": "code",
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Move disk #1 from needle1 to needle3\n",
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"Move disk #2 from needle1 to needle2\n",
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"Move disk #1 from needle3 to needle2\n",
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"Move disk #3 from needle1 to needle3\n",
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"Move disk #1 from needle2 to needle1\n",
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"Move disk #2 from needle2 to needle3\n",
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"Move disk #1 from needle1 to needle3\n"
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]
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}
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],
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"outputs": [],
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"source": [
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"moveDisks(3, 'needle1', 'needle2', 'needle3')"
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]
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"\n",
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"- the following Kattis problems can be solved using recursion:\n",
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"\n",
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"1. Watch Out For Those Hailstones! - https://open.kattis.com/problems/hailstone\n",
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"2. Of of Sorts - https://open.kattis.com/problems/outofsorts\n",
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"1. Last Factorial Digit: https://open.kattis.com/problems/lastfactorialdigit\n",
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"2. Watch Out For Those Hailstones! - https://open.kattis.com/problems/hailstone\n",
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"3. Of of Sorts - https://open.kattis.com/problems/outofsorts\n",
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" - Hint: implement recursive binary search and apply to the generated unsorted sequence"
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]
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},

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